2. If we were to use hill-climbing to minimize an error function f(x,y), we might evaluate the function at some nearby points to the current guess (x0,y{)), then choose the smallest value, and repeat. Of course, if a function is differentiable, then we could do something faster: find the gradient and adjust in the opposite direction. A kind of hybrid approach might be to nd directional derivatives in the ' . /z \\/f/2 directions: [2] a [a] . LIE/2] ' /Z l This effectively samples eight equally spaced possible directions to adjust the location L0] Since the directional derivative Just changes Sign in the opposite direction. (This is definitely not better than gradient descent but we are just curious) For each function they), at the given location (me): 1. Find these four directional derivatives, and note which of the 8 directions has the most negative slope. 4*2=8 points 2. Find the gradient of the function, and its negative. 4*2=8 points 3. Find the difference in f between the point that is 1 unit away in the best of the 8 directions (the one with the most negative slope), and the point that is 1 unit away in the direction of the negative gradient. If the two directions are identical, you can report 0 without evaluating fat the shared location. 2*2 = 4 points i. f(x,y)=xy2+100 at (1,2) ii. f(x,y)=x3y3x2y+l at (1,5)